DMT Duosharp might be a good option. They are pricey but a quality product. Look for monocrystalline vs. polycrystalline. DMT makes a dual sided whetstone that would save you some money. The pruners will not need as fine an edge as your kitchen knife. Sometimes even the small paddle type sharpeners are good to have. They are small and can get into little places without having to take apart the pruners.

Amazon is a good place to get the sharpeners.

Monocrystaline. OK.

Grit size? 200? 400? 600? Fine? Coarse? Medium?

That's the way Amazon has them advertised.

Rain2Fall